Deterministic Chaos

The Logistic Map

The Logistic Map is a model for the growth of a single-species population having non-overlapping generations (for instance children are born in the spring and by next spring are mature and productive - some insect populations are examples), and living in an environment having limited resources. Limited resources enters the model as a competition term: individuals must compete for available food.

How do we build this model? Let P_n stand for the population in generation n:

P₀ is the initial population, the size of the population when we start observng it.

P₁ is the population in the first generation after we start observing.

P₂ is the population in the second generation after we start observing.

and so on.

We would like to find a relation between P_n and P_n+1. If we find this relation, then knowing the population in any generation we can determine the population in all successive generations. What do we know?

First, how would the model look if we ignore competition?

How can we include competition between pairs of animals?

Now we rescale this model to the more familiar form.

Here is some basic geometry of the logistic map.

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